8
6
(K8a
10
)
A knot diagram
1
Linearized knot diagam
5 7 8 6 1 4 3 2
Solving Sequence
1,6
5 2 4 7 8 3
c
5
c
1
c
4
c
6
c
8
c
3
c
2
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1i
* 1 irreducible components of dim
C
= 0, with total 11 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= hu
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1i
(i) Arc colorings
a
1
=
0
u
a
6
=
1
0
a
5
=
1
u
2
a
2
=
u
u
3
+ u
a
4
=
u
2
+ 1
u
2
a
7
=
u
4
+ u
2
+ 1
u
4
a
8
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
10
− u
8
− 2u
6
− u
4
+ u
2
+ 1
−u
10
+ u
9
− u
8
+ 2u
7
− 2u
6
+ 3u
5
− u
4
+ 4u
3
+ u
2
+ u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
10
− 4u
8
− 4u
7
− 12u
6
− 4u
5
− 8u
4
− 8u
3
− 8u
2
− 8u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1
c
2
, c
3
, c
7
u
11
− u
10
− 4u
9
+ 3u
8
+ 6u
7
− 2u
6
− 2u
5
− 3u
4
− 3u
3
+ 3u
2
+ 2u + 1
c
4
, c
6
, c
8
u
11
+ 3u
10
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
11
+ 3y
10
+ ··· − 2y −1
c
2
, c
3
, c
7
y
11
− 9y
10
+ ··· − 2y −1
c
4
, c
6
, c
8
y
11
+ 11y
10
+ ··· + 6y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.274458 + 0.988557I
−5.18162 − 2.94672I −9.79937 + 4.11787I
u = −0.274458 − 0.988557I
−5.18162 + 2.94672I −9.79937 − 4.11787I
u = 0.838197 + 0.796762I
1.97705 − 1.41699I −3.20869 + 0.63373I
u = 0.838197 − 0.796762I
1.97705 + 1.41699I −3.20869 − 0.63373I
u = −0.813506 + 0.895281I
5.64260 − 3.04152I 0.06121 + 2.82242I
u = −0.813506 − 0.895281I
5.64260 + 3.04152I 0.06121 − 2.82242I
u = 0.783273 + 0.973706I
1.43178 + 7.47524I −4.22908 − 5.55460I
u = 0.783273 − 0.973706I
1.43178 − 7.47524I −4.22908 + 5.55460I
u = 0.267638 + 0.666716I
−0.304732 + 1.131300I −4.01220 − 6.05785I
u = 0.267638 − 0.666716I
−0.304732 − 1.131300I −4.01220 + 6.05785I
u = −0.602288
−2.19537 −3.62370
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1
c
2
, c
3
, c
7
u
11
− u
10
− 4u
9
+ 3u
8
+ 6u
7
− 2u
6
− 2u
5
− 3u
4
− 3u
3
+ 3u
2
+ 2u + 1
c
4
, c
6
, c
8
u
11
+ 3u
10
+ ··· − 2u − 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
11
+ 3y
10
+ ··· − 2y −1
c
2
, c
3
, c
7
y
11
− 9y
10
+ ··· − 2y −1
c
4
, c
6
, c
8
y
11
+ 11y
10
+ ··· + 6y −1
7
